The mapping class group power-quotient conjecture for hierarchically hyperbolic groups

Let $S$ be any surface of finite type, and let $g_1,\dots,g_l\in \mathcal{MCG}^\pm(S)$. For a subgroup $H$ of a group $G$, write $\langle\langle H\rangle\rangle$ for its normal closure. Power-quotient conjecture. There exists $N\in\mathbb{N}-\{0\}$ such that, for all $K_1,\dots,K_l\in \mathbb{Z}-\{0\}$,

$$\mathcal{MCG}^\pm(S)/\langle\langle \{g_i^{K_iN}\}\rangle\rangle$$

is hierarchically hyperbolic. The conjecture is open already for quotients by suitable powers of non-separating Dehn twists. It is motivated by the prospect that proving it would lead to a more complete theory of Dehn fillings for hierarchically hyperbolic groups.